REDUCTION OF VULGAR AND DECIMAL FRACTIONS. The value of a vulgar fraction is the quotient arising from dividing the numerator by the denominator. Therefore, CASE 1st.-TO REDUCE A VULGAR FRACTION TO A DECIMAL. RULE. Annex cyphers to the numerator, and divide it by the denominator. Note.-If the given fraction be proper, the quotient will always be a decimal, and will consist of figures equal in number to the cyphers annexed; or, if the number of figures be less, cyphers must be prefixed to complete the number. CASE 2d. TO REDUCE A DECIMAL TO A VULGAR FRACTION. RULE. Write down the given decimal as a numerator, and for a denominator, write 1 with as many cyphers annexed as there are figures in the numerator, and then reduce the fraction to its lowest terms. (See remarks introductory to Decimals.) 1. Reduce the decimal .125 to a vulgar fraction. Performed, 2, (see Case 1st, Vulgar Fractions,) 125-5 25 125 1000' 200, and again, 25÷25=1, Ans. 200 2. Reduce .75 to a vulgar fraction. Performed, 75÷25 =4, Ans. 3. Reduce .9385 to a vulgar fraction. 4. Reduce .2 to a vulgar fraction. 5. Reduce .16 to a vulgar fraction. 6. Reduce .25 to a vulgar fraction. 7. Reduce .45 to a vulgar fraction. 8. Reduce.55 to a vulgar fraction. 9. Reduce .8 to a vulgar fraction. 10. Reduce .24 to a vulgar fraction. 11. Reduce .945 to a vulgar fraction. 12. Reduce .844 to a vulgar fraction. Ans. 18. 20 Ans. 200 250 CASE 3d.-TO REDUCE LOWER DENOMINATIONS TO DECIMALS OF A HIGHER. RULE 1st. Write down the several denominations which are to be reduced to decimals of a higher denomination, one above another, with the lowest uppermost; then divide each denomination, commencing with the lowest, by that number which is required of each to make a unit of the next higher denomination, and at each division place the quotient as a decimal on the right of the next higher denomination. The number last obtained will be the required decimal. Note.-It will be obvious that the division of the lowest denomination must be effected by adding cyphers to that denomination. Cyphers must also be added to each higher denomination to reduce them, unless the decimal figures previously obtained be sufficient. The reason of the above rule is readily shown. Suppose it is required to reduce 7 pence to the fraction of a shilling. The fraction would be, because the shilling is divided into 12 equal parts, and 7 of these parts are taken, and this vulgar fraction is reduced to a decimal by adding cyphers to its numerator and dividing by its denominator. (See Case 1st of Decimals.) Ex. 1. Reduce 7 s. 6 d. to the decimal of a pound sterling. The statement would be, 12/6 That is, the 6 d. is to be divi ded by 12, and the 7 s. by 20. This must be effected by adding cyphers. 2. Reduce 9 d. 2 qr. to the decimal of a pound sterling. RULE 2d. Reduce the given quantity to its lowest denomination, and divide it by a unit of the denomination of the required fraction, reduced to the same denomination. Ex. 3. Reduce 10 s. 9 d. 2 qr. to the decimal of a pound sterling. 10 s. 9 d. 2 qr.=518 qr.; and by the rule, 518 qr. are to be divided by 1 £. reduced to qr. viz. [by 960 qr. Therefore, ៖៖៖ is the fractional answer, and 518-960= .539583+, Ans. 518 To understand the above operation the scholar should remember that 10 s. 9 d. 2 qr. or 518 qr. are to be divided into as many equal parts as there are farthings in 1 £.=960 qr., and one of these parts, 518 or the decimal .539583.+ 4. Reduce 9 s. 8 d. to the decimal of a pound sterling. Ans. .4833.+ 5. Reduce 3 qr. 16 lb. to the decimal of a cwt.? Ans. .8928571.+ 6. Reduce 16 £. 12 s. 8 d. to a decimal expression. Ans. 16.633333.+ 7. Reduce 3 qr. 2 na. to the decimal of a yard. Ans. .875. 8. Reduce 2 roods and 20 rods to the decimal of an acre. Ans. .625. 9. Reduce 3 furlongs 16 rods to the decimal of a mile. Ans. .425. 10. Reduce 12 hours, 15 minutes, and 30 seconds to the decimal of a day. Ans. .51076.+ 11. Reduce 2 cwt. 3 qr. 24 lb. to the decimal of a ton. Ans. .14821428. CASE 4th.-TO FIND THE VALUE OF A DECIMAL IN INTEGERS OF LOWER DENOMINATIONS. RULE.-Multiply the decimal by the number required to reduce it to the next lower denomination, and from the right hand of the product cut off as many figures as there are in the given decimal. The figures on the left of the point will be integers of the denomination next below that given. Proceed in the same way through all the denominations, and the figures on the left of the several points will be the answer required. This rule being directly the reverse of the preceding, needs no explanation. Ex. 1. What is the value of the decimal .5638 of a pound sterling? OPERATION. .5 6 3 8 20. 1 1.2 7 6 0 the shillings and decimals of a shilling in .5638 of a pound sterling. 1 2 3.3 1 2 0 1.2 4 8 0 the pence and decimals of a penny in .2760 of a shilling. the farthings and decimals of a farthing in .3120 of a penny. The value of the above decimal in shillings, pence, &c. is 11 s. 3 d. 1.2480 qr. Note. The integers on the left of the points are the numbers which compose the answer. 2. What is the value of .75 of a pound sterling? Ans. 15s. 3. What is the value of .53854 of a pound sterling? Ans. 10 s. 9 d. 1 qr. nearly. 4. What is the value of .625 of an acre? rods. Ans. 2 roods, 20 5. What is the value of .148712678 of a ton? 3 qr. 25 lb. 1 oz.+ Ans. 2 cwt. 6. What is the value of .676 of a cwt.? Ans. 2 qr. 19 lb. 11 oz.+ 7. How many furlongs, &c. in .425 of a mile? Ans. 3 fur. 16 rods. 8. How many quarters, &c. in .66 of a yard? Ans. 2 qr. 2 nails, 1.26 inches. 9. How many roods, &c. in .321 of an acre? 11 rods, 10 yd. 8 feet. Ans. 1 rood, 10. What is the value of .875 of a hogshead of wine? Ans. 55 gal. 1 pint. 11. What is the value of .875 of a yard? 12. What is the value of .9 of an acre? rods. Ans. 3 qr. 2 nails. QUESTIONS.-How have we contemplated the unit as divided in the preceding rule? How are we to regard it as divided in this rule? In what ratio do the several divisions in decimals decrease? What are they called? What relation is there between integers and decimals? What is the only important consideration which has not already been explained? How many terms are required to express decimals? How may that term be regarded? Repeat the table of whole numbers and decimals. Of what does the denominator of decimal fractions always consist? How do cyphers added to the right of a decimal, affect its value? Give the illustration. How do cyphers placed on the left of a decimal affect its value? Give the illustration. How do decimals and the denominations of federal money correspond? How do we reduce dollars to cents? How do we reduce them to mills? How do we reduce cents and mills to dollars? What is the rule for the addition of decimals? What is it for subtracting decimals? What is the rule for multiplying decimals? How does division stand related to multiplication? What is the rule for division of decimals? What is the rule for pointing off decimals in addition of decimals? What in subtraction? What in multiplication? And what in division? On what does the value of a fraction depend? What is the rule for reducing vulgar fractions to decimals? What is Case 2d? What is the rule for it? What is Case 3d? What is the rule? What note follows the rule? Explain the nature of the rule. What is the second rule? What is Case 4th? What is the rule for it? What note follows the rule? |